Problem 12 - Thinking Mathematically

Question

Construct a truth table for the given statement. \(r \rightarrow(p \vee q)\)

Step-by-Step Solution

Answer

The truth table for the logical statement \(r \rightarrow (p \vee q)\) is as follows: \n\n\[\begin{array}{cccc}r & p & q & r \rightarrow (p \vee q) \\midruleT & T & T & T \T & T & F & T \T & F & T & T \T & F & F & F \F & T & T & T \F & T & F & T \F & F & T & T \F & F & F & T \\end{array}\]

Step 1: Construct the Truth Table for \(r\), \(p\) and \(q\)

First, we construct a table with columns for \(r\), \(p\), \(q\) and all their possible combinations of truth values. Since there are three variables, there will be \(2^3 = 8\) rows.

Step 2: Fill in the Truth Values for the Disjunction \(p \vee q\)

Next, we create a column for the disjunction \(p \vee q\). The disjunction is true if either \(p\) is true, \(q\) is true, or both \(p\) and \(q\) are true.

Step 3: Fill in the Truth Values for \(r \rightarrow (p \vee q)\)

Finally, we create a column for the entire conditional statement \(r \rightarrow (p \vee q)\). Remember that the conditional statement is false if and only if \(r\) is true and \(p \vee q\) is false. In all other cases, the conditional statement is true.

Key Concepts

Logical Operators

Logical operators are a fundamental concept in logic and computer science. They are symbols or words used to connect abstract statements or propositions. These operators help determine the logical relationship and truth values of combined statements. The most common logical operators include:

• AND (\(\land\)): A conjunction that is true only if both combined statements are true.
• OR (\(\lor\)): A disjunction that is true if at least one of the statements is true.
• NOT (\(eg\)): Negates the truth value of a statement, turning true to false, and vice versa.
• IMPLIES (\(\rightarrow\)): Represents a conditional statement showing that one statement's truth leads to the truth of another statement.

Logical operators form the basis of constructing and evaluating truth tables, such as determining the truth value of the expression \(r \rightarrow (p \lor q)\). Each operator has specific rules that govern how the truth values of propositions change when operators are applied. Understanding these rules is crucial for mastering logical reasoning.

Disjunction

A disjunction is a type of logical operator also known as "logical OR." A disjunction combines two propositions, and its primary function is to assert that at least one of the propositions must be true.
For example, in the expression \(p \lor q\), the disjunction \(\lor\) stands for "either or." This means that for the entire expression to be true:

• \(p\) must be true,
• \(q\) must be true, or
• both \(p\) and \(q\) must be true.

Even if one of the propositions is false, the disjunction can still be true as long as one is true. It's significant in the creation of truth tables, where a disjunction's column examines all combinations of truth values for the involved variables. In the given exercise, the disjunction \(p \lor q\) determines part of the truth value for compounds like \(r \rightarrow (p \lor q)\).

Conditional Statement

A conditional statement is a logical construct that asserts truth under certain conditions. It is often expressed in the form "if...then." The logical symbol for implication or conditional statements is \(\rightarrow\).
In a conditional statement \(r \rightarrow (p \lor q)\):

• \(r\) is the hypothesis or antecedent,
• \((p \lor q)\) is the conclusion or consequent.

The truth of a conditional statement depends on both the truth of the antecedent and the consequent.- If \(r\) is true but \((p \lor q)\) is false, then the conditional statement is false.- In all other scenarios, it is considered true.
This rule means that a conditional statement can be true even when the antecedent is false, irrespective of the truth of the consequent. This can initially seem counterintuitive, but it aligns with logical conventions used to construct a truth table.

Truth Values

Truth values are used to determine whether a statement is true or false. They are essential for evaluating logical propositions, statements, and expressions.
Every proposition has two possible truth values:

• True, sometimes represented as 1, "T," or a higher logical designation.
• False, sometimes represented as 0, "F," or a lower logical designation.

Truth tables, like the one constructed for \(r \rightarrow (p \lor q)\), help visually display how truth values interact under various logical operators. Each variable's truth values are listed for every possible combination, helping deduce the compound statement's overall truth.
Understanding how to manipulate truth values with different logical operators allows one to systematically evaluate and validate logical expressions, an essential skill in logic, mathematics, and computer science.